Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation cos(x)-1=0
-
Equation (x^2+4)/x=0
-
Equation 4x+9=5(2x-7)-6x
-
Equation 2*x^2-x-6=0
- Express {x} in terms of y where:
- xy^1=y+x*sin(y/x)
- sqrt(x)*sin(y/x)=0
- 12*10^(-30)-0,1884*10^12*x^2-y(0,3768*10^12*x+1,183152*10^12*x^3)+y^2(2,760688*10^24*x^2)=0=0
- (x^2+y^2)×(x+y-3)=2xy
- The double integral of:
- x-2
- Derivative of:
- x-2
- Graphing y =:
- x-2
-
Identical expressions
- Sqr two x- one =x-2
- Sqr2x minus 1 equally x minus 2
- Sqr two x minus one equally x minus 2
-
Similar expressions
- Sqr2x+1=x-2
- log((9^x)-1,2)+log((9^x)-2,2)=1
- Sqr2x-1=x+2
Sqr2x-1=x-2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(2 x\right)^{2} - 1 = x - 2$$
to
$$\left(2 - x\right) + \left(\left(2 x\right)^{2} - 1\right) = 0$$
Expand the expression in the equation
$$\left(2 - x\right) + \left(\left(2 x\right)^{2} - 1\right) = 0$$
We get the quadratic equation
$$4 x^{2} - x + 1 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 4$$
$$b = -1$$
$$c = 1$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{1}{8} + \frac{\sqrt{15} i}{8}$$
$$x_{2} = \frac{1}{8} - \frac{\sqrt{15} i}{8}$$
left part with negative sign.
The equation is transformed from
$$\left(2 x\right)^{2} - 1 = x - 2$$
to
$$\left(2 - x\right) + \left(\left(2 x\right)^{2} - 1\right) = 0$$
Expand the expression in the equation
$$\left(2 - x\right) + \left(\left(2 x\right)^{2} - 1\right) = 0$$
We get the quadratic equation
$$4 x^{2} - x + 1 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 4$$
$$b = -1$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (4) * (1) = -15
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{1}{8} + \frac{\sqrt{15} i}{8}$$
$$x_{2} = \frac{1}{8} - \frac{\sqrt{15} i}{8}$$
Vieta's Theorem
rewrite the equation
$$\left(2 x\right)^{2} - 1 = x - 2$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{x}{4} + \frac{1}{4} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{4}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{4}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{4}$$
$$x_{1} x_{2} = \frac{1}{4}$$
$$\left(2 x\right)^{2} - 1 = x - 2$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{x}{4} + \frac{1}{4} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{4}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{4}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{4}$$
$$x_{1} x_{2} = \frac{1}{4}$$
Sum and product of roots
[src]
sum
____ ____ 1 I*\/ 15 1 I*\/ 15 - - -------- + - + -------- 8 8 8 8
$$\left(\frac{1}{8} - \frac{\sqrt{15} i}{8}\right) + \left(\frac{1}{8} + \frac{\sqrt{15} i}{8}\right)$$
=
1/4
$$\frac{1}{4}$$
product
/ ____\ / ____\ |1 I*\/ 15 | |1 I*\/ 15 | |- - --------|*|- + --------| \8 8 / \8 8 /
$$\left(\frac{1}{8} - \frac{\sqrt{15} i}{8}\right) \left(\frac{1}{8} + \frac{\sqrt{15} i}{8}\right)$$
=
1/4
$$\frac{1}{4}$$
1/4
Rapid solution
[src]
____
1 I*\/ 15
x1 = - - --------
8 8
$$x_{1} = \frac{1}{8} - \frac{\sqrt{15} i}{8}$$
____
1 I*\/ 15
x2 = - + --------
8 8
$$x_{2} = \frac{1}{8} + \frac{\sqrt{15} i}{8}$$
x2 = 1/8 + sqrt(15)*i/8
Numerical answer
[src]
x1 = 0.125 - 0.484122918275927*i
x2 = 0.125 + 0.484122918275927*i
x2 = 0.125 + 0.484122918275927*i